© 1998 by London Mathematical Society
© The London Mathematical Society
Hammocks and the Nazarova–Roiter Algorithm
Department of Mathematics, Xiamen University 361005 Xiamen, China
Received 7 September 1995.
Hammocks have been considered by Brenner [1], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation-finite algebra. Ringel and Vossieck [11] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of S-spaces, where S is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [8], and it reduces a poset S with a maximal element a to a new poset S'=a
S. The second algorithm was developed by Zavadskij [13], and it reduces a poset S with a suitable pair (a, b) of elements a, b to a new poset S'=(a,b)S. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [5], we discuss the relationship between hammocks and the Zavadskij algorithm.